The James Webb Space Telescope (JWST) will be a nearly ideal machine for acquiring the transmission and emission spectra of transiting exoplanets over its large wavelength range 0.7 - 28 microns. The NIRSpec, NIRCam, nTFI, and MIRI instruments will have spectroscopic capabilities that span spectral resolutions from 20 - 3000 and can cover up to 2 - 3 octaves in wavelength simultaneously. This will allow observing multiple molecular features at once, facilitating the separation of atmospheric temperature and abundance effects on spectra. Many transiting planets will also be able to be observed with both transmission and eclipse spectroscopy, providing further insights and constraints on planetary thermal structures and energy transport. Simulated JWST spectra of planets ranging from mini-Neptunes to gas giants will be presented. These simulations include planets ranging from mini-Neptunes to gas giants will be presented. These simulations include current best estimates of actual instrument throughput, resolution, spectral range, systematic noise, and random noise terms. They show that JWST will be able to determine the atmospheric parameters of a wide variety of planets, often when observing only one or a few transit or eclipse event sequences. The thermal emissions of rocky super-Earths will also be quickly detectable via mid-IR eclipse observations if such planets are found around nearby M star hosts beforehand.